Thread: Triple Double View Single Post
2007-06-01, 08:24   #7
robert44444uk

Jun 2003
Oxford, UK

2·991 Posts

Quote:
 Originally Posted by m_f_h Congrats! What an example of perseverance ! (For novices it would be nice to know an explicit formula linking the small numbers to the big number....) Also, nice Cunningham chain of length 6 - and you would have qualified even by counting it only as 1! Retrospectively, wouldn't that have been an efficient sieving criterion ? (By requiring to have a chain of length 6, you can advance in steps of 6 when scanning n-values. But I don't know if one could expect a candidate with the chain [2,3,4,5,6,7] - what's the relative frequency of such among "good candidates" ?)
For some reason, never satisfactorily explained, twins are less common than CCs, evidenced by the records of 15 twins and 18 CCs. So tacking twins is the most important thing to do to find a triple double.

It is fiendishly difficult to find long bitwin chains, but you are right it is not a bad way to try to find a good base on which to build hoping that it would also provide twins etc. at higher n. In fact I tried this approach, but using mods to build such k tends to overlook the fact that at the next few levels of n, then the values will almost all be composite, which is why I went for a more balanced approach of finding k's with a good number twins in the first few n, and then testing those further

After the t12 discovery, I checked a great number of t9s, but no triple double.

Regarding frequencies:

1. t9 double payams occur with a frequency of 1 in every 1,875,000 odd k

2. t9 double payams that have no factors smaller than 257 in the first 7 n, plus or minus are 1 in 10,500 t9 double payams

3. In a test run of the first 53,256,000 t9s with the property in 2, the probabilities of getting 4,5,6,7 twins in the first 7 n are:

4 twins - 1 in every 1,667 tested
5 twins - 1 in every 37,086 tested
6 twins - 1 in every 873,052 tested
all 7 twin - only one found in that range.

Obviously for higher k, these values drop off quickly.

Hope this helps

Last fiddled with by robert44444uk on 2007-06-01 at 08:28